This is unlikely to be NP-complete, as it can be solved in coRP, using a few calls to PIT. (It follows that this problem is not NP-complete unless $\mathsf{NP} = \mathsf{RP}$ and $\mathsf{PH} = \mathsf{BPP}$.)
I know of two ways to do this:
(1) Using the downward-self-reducibility of the permanent. (From Kabanets-Impagliazzo.) This goes as follows. The Laplace expansion of $perm_n(X)$ is $perm_n(X) = \sum_{i=1}^n x_{1i} perm_{n-1}(X(\hat{1}|\hat{i}))$, where the latter notation means the $(n-1) \times (n-1)$ submatrix that results from removing the first row and the $i$-th column. Next, if $C$ computes the permanent of an $n \times n$ matrix $X$, then $C(\begin{bmatrix} X' & 0_{n \times 1} \\ 0_{1 \times n} & 1 \end{bmatrix}) = perm_{n-1}(X')$. Let us denote this circuit, which has $(n-1)^2$ inputs, as $C_{n-1}$. Then we check the polynomial identity
$C(X) = \sum_{i=1}^n x_{1i} C_{n-1}(X(\hat{1} | \hat{i})$
Then you iterate this for $C_{n-1}$ and so on. That is $n$ polynomial identity tests in total, each of which can be solved in $\mathsf{coRP}$ by the usual algorithm (plug in random values and evaluate).
(2) Using the characterization of the permanent by its symmetries (from Mulmuley, "Explicit proofs and the flip", Prop 7.1). That is, the permanent is the unique polynomial $f(X)$ in $n^2$ variables such that (1) $f(Id)=1$ and (2) $f(P X P^T) = f(X)$ for all permutation matrices $P$, and (3) $f(DXD')=\mu(D,D')f(X)$ for all diagonal matrices $D,D'$, where $\mu(D,D')$ is the product of all their diagonal entries. Since the group of permutation matrices is generated by just two elements, corresponding to the permutations $(12)$ and $(1,2,3,\dotsc,n)$, say, we need only check those two. Then we use an idea similar to the above, as follows. Given a circuit $C(X)$, and a permutation $\pi \in S_n$, let $C_{\pi}$ denote the circuit you get by first permuting the variables $X$ to $P_{\pi} X P_{\pi}^T$ (that is, you send $x_{ij}$ to $x_{\pi(i), \pi(j)}$), and then applying $C$. Then we check that $C(Id)=1$, and we use PIT to check that $C(X) = C_{\pi}(X)$ for $\pi \in \{(12), (1,2,3,\dotsc,n)\}$.
To check the condition about diagonal matrices, we introduce $2n$ new variables corresponding to the entries of those diagonal matrices, and then check the polynomial identity $C(DXD') = \mu(D,D')C(X)$.